Note that ωn = nπc/L, now because &hbar; is so small, we can approxiamate this sum to an integral. In the process we will change the coordinates of the integral over n in spherical coordinates, and we will let x = π&hbar;cn/LT (an extra 1/8 comes in becaues we are integrating over only positive values of n, and an extra 2 due to two independent set of cavity modes of frequencies):

Note: actually, this is a density of states problem with D(n) = 4n2 because of the spherical shell * 1/8 * 2 = n2, ε=&hbar;ωn, and f(ε)=(exp(&hbar;ωn/T) - 1)-1

U = (L3T4/π2&hbar;3c3) ∫0∞ x3/(exp(x) - 1) dx

The integral has a definite value found in an integral table, L3=V, and thus we come upon the stefan-Boltzmann law of radiation: